A generalization of Nash bargaining and proportional fairness to log-convex utility sets with power constraints

Many solutions and concepts in resource allocation and game theory rely on the assumption of a convex utility set. In this paper, we show that the less restrictive assumption of a logarithmic hidden convexity is sometimes sufficient. We consider the problems of Nash bargaining and proportional fairness, which are closely related. We extend the Nash bargaining framework to a broader family of log-convex sets. We then focus on the set of feasible signal-to-interference-plus-noise ratios (SINRs), for the cases of individual power constraints and a sum power constraint. Under the assumption of log-convex interference functions, we show how Pareto optimality of boundary points depends on the interference coupling between the users. Finally, we provide necessary and sufficient conditions for strict log-convexity of the feasible SINR region.